3.1499 \(\int \frac{1}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{b (a+b x)} \]

[Out]

-(1/(b*(a + b*x)))

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Rubi [A]  time = 0.0104686, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{1}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(-1),x]

[Out]

-(1/(b*(a + b*x)))

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Rubi in Sympy [A]  time = 1.89206, size = 29, normalized size = 2.42 \[ - \frac{2 a + 2 b x}{2 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

-(2*a + 2*b*x)/(2*b*(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.00476487, size = 12, normalized size = 1. \[ -\frac{1}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-1),x]

[Out]

-(1/(b*(a + b*x)))

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Maple [A]  time = 0.005, size = 13, normalized size = 1.1 \[ -{\frac{1}{b \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

-1/b/(b*x+a)

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Maxima [A]  time = 0.683971, size = 18, normalized size = 1.5 \[ -\frac{1}{b^{2} x + a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")

[Out]

-1/(b^2*x + a*b)

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Fricas [A]  time = 0.219626, size = 18, normalized size = 1.5 \[ -\frac{1}{b^{2} x + a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")

[Out]

-1/(b^2*x + a*b)

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Sympy [A]  time = 1.1022, size = 10, normalized size = 0.83 \[ - \frac{1}{a b + b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

-1/(a*b + b**2*x)

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GIAC/XCAS [A]  time = 0.208256, size = 16, normalized size = 1.33 \[ -\frac{1}{{\left (b x + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")

[Out]

-1/((b*x + a)*b)